Let $f(x)=2x^4+x^3+x^2-3x+r$. For what value of $r$ is $f(2)=0$?
Evaluating gives  \[f(2)=2(2)^4+(2)^3+(2)^2-3(2)+r=32+8+4-6+r=38+r.\]This is equal to 0 when $r=\boxed{-38}$.